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Absolute Value and Inequalities: Exploring Distance and Tolerance

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Understand absolute value concepts and how to solve equations and inequalities related to distance and tolerance. Explore real number properties and practical examples.

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Absolute Value and Inequalities: Exploring Distance and Tolerance

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  1. Chapter 1.8 Absolute Value and Inequalities

  2. Recall from Chapter R that the absolute value of a number a, written |a|, gives the distance from a to 0 on a number line. By this definition, the equation |x| = 3 can be solved by finding all real numbers at a distance of 3 units from 0.

  3. |x| = 3 Two numbers satisfy this equation, 3 and -3, so the solution set is {-3, 3}. Distance is 3. Distance is 3. Distance is greater than 3. Distance is less than 3. Distance is greater than 3. | | | 3 -3 0

  4. Similarly |x| < 3 is satisfied by all real numbers whose distances from 0 are less than 3, that is, the interval -3<x<3 or (-3,3).

  5. Properties of Absolute Value

  6. Example 1 Solving Absolute Value Equations

  7. Example 1 Solving Absolute Value Equations

  8. Example 2 Solving Absolute Value Inequalities

  9. Example 2 Solving Absolute Value Inequalities

  10. Example 3 Solving Absolute Value Inequalities Requiring a Transformation

  11. Example 4 Solving Absolute Value Equations and Inequalities

  12. Example 4 Solving Absolute Value Equations and Inequalities

  13. Absolute Value Models for Distance and Tolerance Recall from Selection R.2 that is a and b represent two real numbers, then the absolute value of their difference, either |a - b| or |b – a|, represent the distance between them. This fact is used to write absolute value equations or inequalities to express distance.

  14. Example 5 Using Absolute Value Inequaliteis to Describe Distances Write each statement using an absolute value inequality. (a) k is no less than 5 units from 8

  15. Example 5 Using Absolute Value Inequaliteis to Describe Distances Write each statement using an absolute value inequality. (b) n is within .001 unit of 6

  16. Example 6 Using Absolute Value to Model Tolerance Suppose y = 2x + 1 and we want y to be within .01 unit of 4. For what values of x will this be true? |y – 4| < .01 |2x + 1 – 4| < .01

  17. Homework Section 1.8 # 1 - 82

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